Universal approximation by translates of fundamental solutions of elliptic equations

Abstract

A. In the present work, we investigate the approximability of solutions of elliptic partial differential equations in a bounded domain Ω by universal series of translates of fundamental solutions of the underlying partial differential operator. The singularities of the fundamental solutions lie on a prescribed surface outside of Ω, known as the pseudo–boundary. The domains under consideration satisfy a rather mild boundary regularity requirement, namely, the Segment Condition. We study approximations with respect to the norms of the spaces C`(Ω) and we establish the existence of universal series. Analogous results are obtainable with respect to the norms of Holder spaces C`,ν(Ω) and Sobolev spaces Wk,p(Ω). The sequence a = {an}n∈N of coefficients of the universal series may be chosen in ∩p>1l(N) but it can not be chosen in l1(N).